I think the bits as basically very low quality. When you go to your favorite store, simply buy one of the contractor grade of bits like DeWalt, Kolbalt, Irwin, Vermont American etc. Be careful what kind of jobs you tackle with your light duty screwdriver. I'm sure it does not have the power to drive longer screws into hard woods like decking etc. You have to hold a lot of force against the screw in these situations or the bit will slip, spin and dull, as well as strip out the screw head.
The Phillips screwdriver size is based on the size at the very tip. Based on diagrams of the tip from the original Phillips patent (shown in one of the links to sizes.com), I'm guessing that the relevant physical size is the cross you see when you point the screwdriver at your nose (as though it were a screw) and look at it end-on, or something directly related to that. How much the tip widens to become part of the shaft is irrelevant. You could make one that widens to a meter in diameter, making it nearly impossible for humans to use.
I tried this experiment: using some digital calipers, insert just the tip between the calipers, and measure the distance from one point on the cross to the opposite point. This is difficult to do accurately since if you insert a little more than the tip, you get a result that is too large. I tried several different sets of screwdrivers and bits for them from different manufacturers.
The results: PH3: 0.141 inches, PH2: 0.086 inches, PH1: 0.071 inches, PH0: 0.055 inches, PH00: 0.048 inches, PH000: 0.039 inches, and PH0000 is so tiny I really couldn't get a consistent reading. I don't have a PH4 and I'm not sure I've seen one. As I said, it's difficult to get an accurate reading. I can quite consistently eyeball the difference between a PH3 and a PH2 or a PH2 and a PH1. It's difficult to consistently see that a PH00 is larger than a PH000. I can see that like sizes give similar results, and measuring the same bit over and over also gives similar results, however, the ranges for adjacent sizes overlap a bit for PH000, PH00, and PH0.
A serious attempt to make these measurements would probably require several vices, a powerful magnifying glass or a microscope, and digital calipers with a thinner edge than the set I have. Either that or I'd need about 6 hands.
I presume a PH0000 screwdriver is intended to drive #0000 screws, a PH000 screwdriver is intended to drive #000 screws, and a PH00 screwdriver is intended to drive #00 screws, as these seem to be the only screws smaller than #0 in the inch-based "numbered screw" system. There could be a #00000 screw, but it's less than one-hundredth of an inch wide and probably can't take much torque, and a #000000 screw would have a negative diameter, which is obviously impossible.
Somewhere buried in an ANSI standard is the physical dimension that determines the driver size. I may have found one of them, but it costs $65.
Best Answer
Look at Table 1 in this reference, it gives the minimum and maximum dimensions for the square based on the screw sizes. The maximum allowed dimensions of the square are as follows (including my conversion to mm):
Drive Max dimension
0 - 0.0710 in 1.8 mm
1 - 0.0910 in 2.31 mm
2 - 0.1126 in 2.86 mm
3 - 0.1330 in 3.38 mm
4 - 0.1910 in 4.85 mm